A073009 -id:A073009 - cp's OEIS frontend

A083649 Duplicate of A073009.

1, 1, 2, 9, 1, 2, 8, 5, 9, 9, 7, 0, 6, 2, 6, 6, 3, 5, 4, 0, 4, 0, 7, 2, 8, 2, 5, 9, 0, 5, 9, 5, 6, 0, 0

Offset: 0

dead

A000312 a(n) = n^n; number of labeled mappings from n points to themselves (endofunctions).

1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10000000000, 285311670611, 8916100448256, 302875106592253, 11112006825558016, 437893890380859375, 18446744073709551616, 827240261886336764177, 39346408075296537575424, 1978419655660313589123979

Offset: 0

N. J. A. Sloane

Also number of labeled pointed rooted trees (or vertebrates) on n nodes.
For n >= 1 a(n) is also the number of n X n (0,1) matrices in which each row contains exactly one entry equal to 1. - Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001
Also the number of labeled rooted trees on (n+1) nodes such that the root is lower than its children. Also the number of alternating labeled rooted ordered trees on (n+1) nodes such that the root is lower than its children. - Cedric Chauve (chauve(AT)lacim.uqam.ca), Mar 27 2002
With p(n) = the number of integer partitions of n, p(i) = the number of parts of the i-th partition of n, d(i) = the number of different parts of the i-th partition of n, p(j, i) = the j-th part of the i-th partition of n, m(i, j) = multiplicity of the j-th part of the i-th partition of n, one has: a(n) = Sum_{i=1..p(n)} (n!/(Product_{j=1..p(i)} p(i, j)!)) * ((n!/(n - p(i)))!/(Product_{j=1..d(i)} m(i, j)!)). - Thomas Wieder, May 18 2005
All rational solutions to the equation x^y = y^x, with x < y, are given by x = A000169(n+1)/A000312(n), y = A000312(n+1)/A007778(n), where n = 1, 2, 3, ... . - Nick Hobson, Nov 30 2006
a(n) is the total number of leaves in all (n+1)^(n-1) trees on {0,1,2,...,n} rooted at 0. For example, with edges directed away from the root, the trees on {0,1,2} are {0->1,0->2},{0->1->2},{0->2->1} and contain a total of a(2)=4 leaves. - David Callan, Feb 01 2007
Limit_{n->infinity} A000169(n+1)/a(n) = exp(1). Convergence is slow, e.g., it takes n > 74 to get one decimal place correct and n > 163 to get two of them. - Alonso del Arte, Jun 20 2011
Also smallest k such that binomial(k, n) is divisible by n^(n-1), n > 0. - Michel Lagneau, Jul 29 2013
For n >= 2 a(n) is represented in base n as "one followed by n zeros". - R. J. Cano, Aug 22 2014
Number of length-n words over the alphabet of n letters. - Joerg Arndt, May 15 2015
Number of prime parking functions of length n+1. - Rui Duarte, Jul 27 2015
The probability density functions p(x, m=q, n=q, mu=1) = A000312(q)*E(x, q, q) and p(x, m=q, n=1, mu=q) = (A000312(q)/A000142(q-1))*x^(q-1)*E(x, q, 1), with q >= 1, lead to this sequence, see A163931, A274181 and A008276. - Johannes W. Meijer, Jun 17 2016
Satisfies Benford's law [Miller, 2015]. - N. J. A. Sloane, Feb 12 2017
A signed version of this sequence apart from the first term (1, -4, -27, 256, 3125, -46656, ...), has the following property: for every prime p == 1 (mod 2n), (-1)^(n(n-1)/2)*n^n = A057077(n)*a(n) is always a 2n-th power residue modulo p. - Jianing Song, Sep 05 2018
From Juhani Heino, May 07 2019: (Start)
n^n is both Sum_{i=0..n} binomial(n,i)*(n-1)^(n-i)
  and Sum_{i=0..n} binomial(n,i)*(n-1)^(n-i)*i.
The former is the familiar binomial distribution of a throw of n n-sided dice, according to how many times a required side appears, 0 to n. The latter is the same but each term is multiplied by its amount. This means that if the bank pays the player 1 token for each die that has the chosen side, it is always a fair game if the player pays 1 token to enter - neither bank nor player wins on average.
Examples:
2-sided dice (2 coins): 4 = 1 + 2 + 1 = 1*0 + 2*1 + 1*2 (0 omitted from now on);
3-sided dice (3 long triangular prisms): 27 = 8 + 12 + 6 + 1 = 12*1 + 6*2 + 1*3;
4-sided dice (4 long square prisms or 4 tetrahedrons): 256 = 81 + 108 + 54 + 12 + 1 = 108*1 + 54*2 + 12*3 + 1*4;
5-sided dice (5 long pentagonal prisms): 3125 = 1024 + 1280 + 640 + 160 + 20 + 1 = 1280*1 + 640*2 + 160*3 + 20*4 + 1*5;
6-sided dice (6 cubes): 46656 = 15625 + 18750 + 9375 + 2500 + 375 + 30 + 1 = 18750*1 + 9375*2 + 2500*3 + 375*4 + 30*5 + 1*6.
(End)
For each n >= 1 there is a graph on a(n) vertices whose largest independent set has size n and whose independent set sequence is constant (specifically, for each k=1,2,...,n, the graph has n^n independent sets of size k). There is no graph of smaller order with this property (Ball et al. 2019). - David Galvin, Jun 13 2019
For n >= 2 and 1 <= k <= n, a(n)*(n + 1)/4 + a(n)*(k - 1)*(n + 1 - k)/2*n is equal to the sum over all words w = w(1)...w(n) of length n over the alphabet {1, 2, ..., n} of the following quantity: Sum_{i=1..w(k)} w(i). Inspired by Problem 12432 in the AMM (see links). - Sela Fried, Dec 10 2023
Also, dimension of the unique cohomology group of the smallest interval containing the poset of partitions decorated by Perm, i.e. the poset of pointed partitions. - Bérénice Delcroix-Oger, Jun 25 2025

			G.f. = 1 + x + 4*x^2 + 27*x^3 + 256*x^4 + 3125*x^5 + 46656*x^6 + 823543*x^7 + ...

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pp. 62, 63, 87.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 173, #39.
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.2.37)
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Kenny Lau, Table of n, a(n) for n = 0..385 [First 100 terms computed by T. D. Noe]
Taylor Ball, David Galvin, Katie Hyry, and Kyle Weingartner, Independent set and matching permutations, arXiv:1901.06579 [math.CO], 2019.
Arthur T. Benjamin and Fritz Juhnke, Another way of counting n^n, SIAM J. Discrete Math., Vol. 5, No. 3 (1992), pp. 377-379. - _N. J. A. Sloane_, Jun 09 2011
H. Bottomley, Illustration of initial terms.
H. J. Brothers and J. A. Knox, New closed-form approximations to the logarithmic constant e, The Mathematical Intelligencer, Vol. 20 (4), 1998, pp. 25-29. (Sequence appears as formula in Eq. (8))
C. Chauve, S. Dulucq and O. Guibert, Enumeration of some labeled trees, Proceedings of FPSAC/SFCA 2000 (Moscow), Springer, pp. 146-157.
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 34.
Frank Ellermann, Illustration of binomial transforms.
José María Grau and Antonio M. Oller-Marcén, On the last digit and the last non-zero digit of n^n in base b, Bulletin of the Korean Mathematical Society, Vol. 51, No. 5 (2014), pp. 1325-1337; arXiv preprint, arXiv:1203.4066 [math.NT], 2012.
Nick Hobson, Solution to puzzle 48: Exponential equation.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 36.
Steven J. Miller (ed.), Exercises to "The Theory and Applications of Benford's Law", Princeton University Press, 2015.
Mustafa Obaid et al., The number of complete exceptional sequences for a Dynkin algebra, arXiv preprint arXiv:1307.7573 [math.RT], 2013.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
E. Vigren (Proposer), Problem 12432, Amer. Math. Monthly 130 (2023), p. 953.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See pp. 12-13.
Eric Weisstein's World of Mathematics, Hadamard's Maximum Determinant Problem.
Eric Weisstein's World of Mathematics, Hankel Matrix.
Dimitri Zvonkine, An algebra of power series..., arXiv:math/0403092 [math.AG], 2004.
Index entries for "core" sequences
Index entries for sequences related to rooted trees
Index entries for sequences related to Benford's law

Cf. A000107, A000169, A000272, A001372, A007778, A007830, A008785-A008791, A019538, A048993, A008279, A085741, A062206, A212333.
First column of triangle A055858. Row sums of A066324.
Cf. A001923 (partial sums), A002109 (partial products), A007781 (first differences), A066588 (sum of digits).
Cf. A056665, A081721, A130293, A168658, A275549-A275558 (various classes of endofunctions).
Cf. A174824, A204688.
Cf. A055137, A083648.

a000312 n = n ^ n
a000312_list = zipWith (^) [0..] [0..]  -- Reinhard Zumkeller, Jul 07 2012

A000312 := n->n^n: seq(A000312(n), n=0..17);

Array[ #^# &, 16] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
Table[Sum[StirlingS2[n, i] i! Binomial[n, i], {i, 0, n}], {n, 0, 20}] (* Geoffrey Critzer, Mar 17 2009 *)
a[ n_] := If[ n < 1, Boole[n == 0], n^n]; (* Michael Somos, May 24 2014 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1 / (1 + LambertW[-x]), {x, 0, n}]]; (* Michael Somos, May 24 2014 *)
a[ n_] := If[n < 0, 0, n! SeriesCoefficient[ Nest[ 1 / (1 - x / (1 - Integrate[#, x])) &, 1 + O[x], n], {x, 0, n}]]; (* Michael Somos, May 24 2014 *)
a[ n_] := If[ n < 0, 0, With[{m = n + 1}, m! SeriesCoefficient[ InverseSeries[ Series[ (x - 1) Log[1 - x], {x, 0, m}]], m]]]; (* Michael Somos, May 24 2014 *)

A000312[n]:=if n=0 then 1 else n^n$
makelist(A000312[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

PARI
```
{a(n) = n^n};
```

is(n)=my(b,k=ispower(n,,&b));if(k,for(e=1,valuation(k,b), if(k/b^e == e, return(1)))); n==1 \\ Charles R Greathouse IV, Jan 14 2013

{a(n) = my(A = 1 + O(x)); if( n<0, 0, for(k=1, n, A = 1 / (1 - x / (1 - intformal( A)))); n! * polcoeff( A, n))}; /* Michael Somos, May 24 2014 */

def A000312(n): return n**n # Chai Wah Wu, Nov 07 2022

a(n-1) = -Sum_{i=1..n} (-1)^i*i*n^(n-1-i)*binomial(n, i). - Yong Kong (ykong(AT)curagen.com), Dec 28 2000
E.g.f.: 1/(1 + W(-x)), W(x) = principal branch of Lambert's function.
a(n) = Sum_{k>=0} binomial(n, k)*Stirling2(n, k)*k! = Sum_{k>=0} A008279(n,k)*A048993(n,k) = Sum_{k>=0} A019538(n,k)*A007318(n,k). - Philippe Deléham, Dec 14 2003
E.g.f.: 1/(1 - T), where T = T(x) is Euler's tree function (see A000169).
a(n) = A000169(n+1)*A128433(n+1,1)/A128434(n+1,1). - Reinhard Zumkeller, Mar 03 2007
Comment on power series with denominators a(n): Let f(x) = 1 + Sum_{n>=1} x^n/n^n. Then as x -> infinity, f(x) ~ exp(x/e)*sqrt(2*Pi*x/e). - Philippe Flajolet, Sep 11 2008
E.g.f.: 1 - exp(W(-x)) with an offset of 1 where W(x) = principal branch of Lambert's function. - Vladimir Kruchinin, Sep 15 2010
a(n) = (n-1)*a(n-1) + Sum_{i=1..n} binomial(n, i)*a(i-1)*a(n-i). - Vladimir Shevelev, Sep 30 2010
With an offset of 1, the e.g.f. is the compositional inverse ((x - 1)*log(1 - x))^(-1) = x + x^2/2! + 4*x^3/3! + 27*x^4/4! + .... - Peter Bala, Dec 09 2011
a(n) = denominator((1 + 1/n)^n) for n > 0. - Jean-François Alcover, Jan 14 2013
a(n) = A089072(n,n) for n > 0. - Reinhard Zumkeller, Mar 18 2013
a(n) = (n-1)^(n-1)*(2*n) + Sum_{i=1..n-2} binomial(n, i)*(i^i*(n-i-1)^(n-i-1)), n > 1, a(0) = 1, a(1) = 1. - Vladimir Kruchinin, Nov 28 2014
log(a(n)) = lim_{k->infinity} k*(n^(1+1/k) - n). - Richard R. Forberg, Feb 04 2015
From Ilya Gutkovskiy, Jun 18 2016: (Start)
Sum_{n>=1} 1/a(n) = 1.291285997... = A073009.
Sum_{n>=1} 1/a(n)^2 = 1.063887103... = A086648.
Sum_{n>=1} n!/a(n) = 1.879853862... = A094082. (End)
A000169(n+1)/a(n) -> e, as n -> oo. - Daniel Suteu, Jul 23 2016
a(n) = n!*Product_{k=1..n} binomial(n, k)/Product_{k=1..n-1} binomial(n-1, k) = n!*A001142(n)/A001142(n-1). - Tony Foster III, Sep 05 2018
a(n-1) = abs(p_n(2-n)), for n > 2, the single local extremum of the n-th row polynomial of A055137 with Bagula's sign convention. - Tom Copeland, Nov 15 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = A083648. - Amiram Eldar, Jun 25 2021
Limit_{n->oo} (a(n+1)/a(n) - a(n)/a(n-1)) = e (see Brothers/Knox link). - Harlan J. Brothers, Oct 24 2021
Conjecture: a(n) = Sum_{i=0..n} A048994(n, i) * A048993(n+i, n) for n >= 0; proved by Mike Earnest, see link at A354797. - Werner Schulte, Jun 19 2022

A083648 Decimal expansion of Sum_{n>=1} -(-1)^n/n^n = Integral_{x=0..1} x^x dx.

Original entry on oeis.org

7, 8, 3, 4, 3, 0, 5, 1, 0, 7, 1, 2, 1, 3, 4, 4, 0, 7, 0, 5, 9, 2, 6, 4, 3, 8, 6, 5, 2, 6, 9, 7, 5, 4, 6, 9, 4, 0, 7, 6, 8, 1, 9, 9, 0, 1, 4, 6, 9, 3, 0, 9, 5, 8, 2, 5, 5, 4, 1, 7, 8, 2, 2, 7, 0, 1, 6, 0, 0, 1, 8, 4, 5, 8, 9, 1, 4, 0, 4, 4, 5, 6, 2, 4, 8, 6, 4, 2, 0, 4, 9, 7, 2, 2, 6, 8, 9, 3, 8, 9, 7, 4, 8, 0, 0

Offset: 0

Eric W. Weisstein, May 01 2003

In 1697, Johann Bernoulli explores this curve and finds its minimum and the area under the curve from 0 to 1, all this without the benefit of the exponential function.

			0.78343051071213440705926438652697546940768199014693095825541782270...

William Dunham, The Calculus Gallery, Masterpieces from Newton to Lebesgue, Princeton University Press, Princeton, NJ 2005, pp. 46-51.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.11, p. 449.
Paul J. Nahin, An Imaginary Tale: The Story of sqrt(-1), Princeton, New Jersey: Princeton University Press (1988), p. 146.

G. C. Greubel, Table of n, a(n) for n = 0..5000
M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.
Eric Weisstein's World of Mathematics, Power Tower.
Eric Weisstein's World of Mathematics, Sophomore's Dream.

Cf. A137420 (continued fraction expansion).

Cf. A073009. The minimum point on the curve x^x is (A068985, A072364).

RealDigits[ Sum[ -(-1)^n /n^n, {n, 1, 60}], 10, 111] [[1]] (* Robert G. Wilson v, Jan 31 2005 *)

-sumalt(n=1, (-1/n)^(n)) \\ Michel Marcus, Oct 15 2015

numerical_approx(-sum((-1/n)^n for n in (1..120)), digits=130) # G. C. Greubel, Mar 01 2019

Constant also equals the double integral Integral_{y = 0..1} Integral_{x = 0..1} (x*y)^(x*y) dx dy. - Peter Bala, Mar 04 2012
From Petros Hadjicostas, Jun 29 2020: (Start)
Equals -Integral_{x=0..1, y=0..1} (x*y)^(x*y)/log(x*y) dx dy. (Apply Theorem 1 or Theorem 2 of Glasser (2019) to Integral_{x=0..1} x^x dx.)
Equals -Integral_{x=0..1} x^x*log(x) dx. (Apply Theorem 1 or Theorem 2 of Glasser (2019) to the double integral of Peter Bala above.)
Without using the results in Glasser (2019), notice that Integral x^x*(1 + log(x)) dx = x^x + c, which implies Integral_{x=0..1} x^x dx = -Integral_{x=0..1} x^x*log(x) dx. (End)

A094289 Decimal expansion of Sum(1/p^p) where p is prime.

Original entry on oeis.org

2, 8, 7, 3, 5, 8, 2, 5, 1, 3, 0, 6, 2, 2, 4, 1, 7, 9, 7, 3, 6, 4, 1, 8, 0, 4, 5, 8, 7, 8, 9, 3, 2, 2, 0, 6, 9, 5, 5, 9, 0, 8, 8, 0, 2, 6, 8, 5, 8, 8, 1, 7, 0, 9, 2, 9, 9, 4, 9, 9, 3, 6, 8, 9, 4, 7, 0, 8, 9, 3, 2, 9, 2, 7, 8, 6, 8, 8, 9, 3, 9, 7, 7, 0, 2, 0, 9, 1, 2, 4, 2, 8, 0, 0, 2, 9, 0, 9, 0, 0, 5, 5, 9, 2, 9

Offset: 0

Cino Hilliard, Jul 31 2004

This constant approximately equals 5226294/18187381. - Yalcin Aktar, Nov 05 2006

The asymptotic mean of A129251. - Amiram Eldar, Nov 07 2022

			0.287358251306224179736418045878932206955908802685881709299499368947089...

Cf. A073009, A083648, A051674 (prime(n)^prime(n)), A129251.

digits = 105; n0 = 10; dn = 10; Clear[f]; f[n_] := f[n] = RealDigits[ Sum[ 1/Prime[k]^Prime[k], {k, 1, n}], 10, digits+5] // First; f[n = n0]; f[n = n+dn]; While[Print["n = ", n]; f[n] != f[n-dn], n = n+dn]; Take[f[n], digits] (* Jean-François Alcover, Nov 22 2013 *)

ptothep(n) = { local(x,s,a); default(realprecision,200); s=0; forprime(x=2,n,s+=1./x^x); a=Vec(Str(s)); for(x=3,n,print1(eval(a[x]),",")) }

Offset corrected by R. J. Mathar, Feb 05 2009

A100085 Decimal expansion of Sum_{n>0} 1/(n!^n!).

Original entry on oeis.org

1, 2, 5, 0, 0, 2, 1, 4, 3, 3, 4, 7, 0, 5, 0, 7, 5, 4, 4, 5, 8, 1, 6, 1, 8, 6, 5, 5, 6, 9, 2, 7, 3, 0, 5, 1, 6, 5, 7, 7, 5, 3, 4, 7, 0, 6, 2, 1, 8, 8, 6, 5, 7, 6, 8, 3, 0, 7, 4, 2, 9, 2, 0, 3, 7, 0, 2, 7, 4, 9, 6, 5, 1, 0, 3, 8, 1, 8, 9, 6, 0, 5, 1, 9, 6, 3, 5, 8, 7, 8, 2, 7, 4, 6, 2, 6, 1, 4, 4, 4, 4, 7, 9

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 08 2004

This number was called the Pomerance Number, after Carl Pomerance, in the paper by Bailey and Crandall referenced here. The paper by Martin contains a suggestion in its the Acknowledgements section by Carl Pomerance that the number might be "absolutely abnormal".

			1.250021433470507544581618655692730516577534706218865768307...

G. Harman, One hundred years of normal numbers, in M. A. Bennett et al., eds., Number Theory for the Millennium, II (Urbana, IL, 2000), 149-166, A K Peters, Natick, MA, 2002.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000.
D. H. Bailey and R. E. Crandall, On the random character of fundamental constant expansions
G. Martin, Absolutely abnormal numbers arXiv:math/0006089 [math.NT], 2000; American Mathematical Monthly 108 (October):746-754.

Cf. A073009, A099870, A099871, A099872, A099873, A100084.

RealDigits[ Sum[1/(n!)^(n!), {n, 4}], 10, 111][[1]] (* Robert G. Wilson v, Feb 26 2008 *)

suminf(n=1, 1/(n!^n!)) \\ Michel Marcus, Dec 22 2016

Edited by N. J. A. Sloane, May 16 2008 at the suggestion of R. J. Mathar

A099870 Decimal expansion of Sum_{n>0} 1/(n^log(n)).

Original entry on oeis.org

2, 2, 3, 8, 1, 8, 1, 3, 0, 6, 7, 9, 6, 6, 9, 3, 0, 4, 3, 1, 8, 3, 1, 3, 6, 9, 9, 4, 1, 9, 9, 7, 1, 8, 0, 0, 9, 6, 1, 6, 1, 8, 1, 0, 8, 1, 7, 6, 5, 0, 0, 5, 4, 2, 2, 3, 9, 1, 5, 9, 0, 5, 0, 8, 1, 1, 6, 8, 2, 6, 9, 2, 7, 4, 6, 6, 2, 7, 0, 1, 2, 7, 7, 5, 7, 0, 5, 6, 4, 8, 4, 8, 3, 5, 3, 5, 5, 8, 1, 0, 8, 0, 1, 8, 6

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Oct 29 2004

This series converges slowly. - Bernard Schott, May 23 2019

This series converges more slowly than Sum_{n>=0} 1/a^n for every a > 1 but faster than Sum_{n>=1} 1/n^p for every p > 1. - Jianing Song, Jul 25 2021

			2.23818130679669304318313699419971800961618108176500542239159050811...

J. M. Arnaudiès, P. Delezoide et H. Fraysse, Exercices résolus d'Analyse du cours de mathématiques - 2, Dunod, Exercice 9c, page 293.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A073009, A099871.

SetDefaultRealField(RealField(100)); [(&+[1/k^Log(k): k in [1..1000]])]; // G. C. Greubel, Nov 20 2018

evalf(sum(1/(n^log(n)), n=1..infinity), 110); \\ Bernard Schott, May 23 2019

s = 0; Do[s = N[s + 1/n^Log[n], 256], {n, 10^7}]; RealDigits[s, 10, 111][[1]] (* Robert G. Wilson v, Nov 02 2004 *)

default(realprecision,35);sum(n=1,50000,1./(n^log(n)))

sumpos(n=1, 1/(n^log(n))) \\ Michel Marcus, May 24 2019

numerical_approx(sum(1/k^log(k) for k in [1..1000]), digits=100) # G. C. Greubel, Nov 20 2018

Equals Sum_{n>=1} 1/(n^log(n)).

More terms from Robert G. Wilson v, Nov 02 2004

A085534 a(n) = (2n)^(2n).

Original entry on oeis.org

1, 4, 256, 46656, 16777216, 10000000000, 8916100448256, 11112006825558016, 18446744073709551616, 39346408075296537575424, 104857600000000000000000000, 341427877364219557396646723584, 1333735776850284124449081472843776, 6156119580207157310796674288400203776

Offset: 0

N. J. A. Sloane, Jul 05 2003

nonn

All terms are both perfect squares and numbers of the form n^n. - William Boyles, Jul 31 2015
Intersection of A000290 and A000312. - Michel Marcus, Aug 04 2015
Intersection of A005843 and A000312. - Robert Israel, Aug 04 2015
The number of sequences of length 2n using 2n symbols. - Washington Bomfim, Jan 14 2020

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A000312, A073009, A083648, A085529.

Column k=0 of A246070.

{1}~Join~Table[(2 n)^(2 n), {n, 1, 4!}] (* Michael De Vlieger, Aug 04 2015 *)

def A085534(n): return (m:=n<<1)**m # Chai Wah Wu, Nov 18 2022

a(n) = A000312(2*n). - Michel Marcus, Jul 31 2015
a(n) = A062971(n)^2. - Michel Marcus, Aug 04 2015
a(n) = [x^(2*n)] 1/(1 - 2*n*x). - Ilya Gutkovskiy, Oct 10 2017
Sum_{n>=0} 1/a(n) = 1 + (A073009-A083648)/2 = 1.2539277431... . - Amiram Eldar, May 17 2022

A098686 Decimal expansion of Sum_{n >= 1} n/(n^n).

Original entry on oeis.org

1, 6, 2, 8, 4, 7, 3, 7, 1, 2, 9, 0, 1, 5, 8, 4, 4, 4, 7, 0, 5, 5, 8, 8, 9, 1, 4, 3, 2, 6, 1, 8, 8, 3, 0, 3, 1, 6, 5, 0, 5, 4, 0, 3, 1, 0, 9, 5, 4, 6, 2, 1, 4, 1, 6, 4, 7, 4, 1, 3, 6, 4, 3, 0, 0, 9, 2, 3, 8, 5, 9, 7, 0, 5, 1, 8, 1, 1, 9, 8, 0, 4, 8, 6, 4, 3, 2, 6, 4, 4, 0, 3, 1, 2, 9, 6, 2, 0, 5, 3, 4, 3, 6, 5, 2

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Oct 27 2004

From Peter Bala, Oct 17 2019: (Start)
Equals 1 + Integral_{x = 0..1} x/x^x dx. More generally, for k = 0,1,2,..., Sum_{n >= k+1} n^k/n^n = Integral_{x = 0..1} x^k/x^x dx.
Also equals the double integral Integral_{x = 0..1, y = 0..1} (1 + x*y)/ (x*y)^(x*y) dx dy. Cf. A073009. (End)
Equals Integral_{x = 0..1} (1 - x*log(x))/x^x dx. - Peter Bala, Jul 21 2022
From Peter Bala, Nov 02 2022: (Start)
Equals Integral_{x = 0..1} (1 + x*log(x)^2)/x^x dx.
Equals the double integral Integral_{x = 0..1, y = 0..1} (x*y*log(x*y) - 1)/( (x*y)^(x*y) * log(x*y) ) dx dy and also equals 1 - Integral_{x = 0..1, y = 0..1} x*y/( (x*y)^(x*y) * log(x*y) ) dx dy by Glasser, Theorem 1. (End)

			1.62847371290158444705588914326188303165054031095462141647413643009...

M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.

Cf. A001113, A013661, A073009.

evalf(add(n/(n^n), n = 0..65), 100); # Peter Bala, Nov 02 2022

s = 0; Do[s = N[s + n/n^n, 128], {n, 62}]; RealDigits[s, 10, 111][[1]] (* Robert G. Wilson v, Nov 03 2004 *)

suminf(n=1, 1/n^(n-1)) \\ Michel Marcus, Oct 21 2019

More terms from Robert G. Wilson v, Nov 03 2004

A134877 Decimal expansion of Sum_{k>=1} 1/(k!!)^(k!!).

Original entry on oeis.org

1, 2, 8, 7, 0, 3, 7, 0, 9, 6, 6, 4, 1, 6, 8, 1, 8, 1, 4, 7, 1, 1, 3, 2, 0, 2, 9, 7, 5, 5, 8, 2, 0, 4, 2, 5, 9, 0, 4, 2, 1, 6, 9, 5, 5, 3, 4, 0, 2, 2, 2, 0, 8, 1, 0, 2, 6, 7, 9, 8, 7, 6, 5, 9, 2, 6, 9, 0, 9, 9, 7, 3, 0, 0, 3, 5, 1, 8, 6, 6, 5, 4, 1, 9, 3, 0, 7, 3, 6, 8, 0, 1, 0, 8, 8, 0, 8, 3, 9, 3, 7, 7, 8, 7, 5

Offset: 1

Artur Jasinski, Nov 14 2007

			Sum_{k>=1} 1/(k!!)^(k!!) = 1.28703709664168181471132029755820425904216955340222081026798765926909...

Cf. A073009, A100085.

RealDigits[N[Sum[1/n^n, {n, 1, 100}], 200]][[1]]

A245637 Decimal expansion of Integral_{x = 1..infinity} 1/x^x dx.

Original entry on oeis.org

7, 0, 4, 1, 6, 9, 9, 6, 0, 4, 3, 7, 4, 7, 4, 4, 6, 0, 0, 1, 1, 4, 4, 2, 1, 0, 7, 8, 5, 7, 1, 2, 3, 8, 1, 0, 5, 8, 7, 5, 9, 7, 2, 6, 8, 6, 9, 3, 4, 5, 6, 5, 5, 5, 4, 7, 8, 2, 9, 7, 6, 1, 5, 8, 4, 6, 0, 8, 7, 0, 7, 8, 3, 8, 1, 3, 3, 1, 9, 0, 7, 5, 0, 8, 1, 3, 7, 8, 8, 6, 6, 6, 0, 0, 3, 4, 1, 6, 8, 0, 7, 3, 1, 7

Offset: 0

Jean-François Alcover, Jul 28 2014

			0.704169960437474460011442107857123810587597268693456555478297615846...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.6 Fransén-Robinson Constant, p. 263.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Peter Bala, Borel summation of a family of divergent series
M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.
Math Overflow, Value of divergent sum Sum_{n >= 0} (-1)^n*n^n.
G. N. Watson, Theorems stated by Ramanujan (VIII): Theorems on Divergent Series, Journal of the London Mathematical Society, Volume s1-4, Issue 2, April 1929, Pages 82-86.

Cf. A073009, A083648, A229191.

NIntegrate[1/x^x, {x, 1, Infinity}, WorkingPrecision -> 104] // RealDigits // First

Equals A229191 - A073009. - Vaclav Kotesovec, Jul 28 2014
From Peter Bala, Nov 10 2019: (Start)
Equals Integral_{x = 1..oo} x*(1 + log(x))/x^x dx - 1.
Equals Integral_{x = 1..oo} x*(1 - log^2(x))/x^x dx.
Conjecturally, equals 1 - Integral_{x = 1..oo, y = 1..oo} 1/(x*y)^(x*y) dx dy. [added Dec 21 2022: follows from Glasser's Theorem 1.] (End)
From Peter Bala, Dec 21 2022: (Start)
Equals 1 - Integral_{x = 1..oo} log(x)/x^x dx (since d/d(1/x^x) = -(1 + log(x))/x^x).
Equals the Borel sum of the divergent series 1 - 1^1 + 2^2 - 3^3 + 4^4 - .... See Watson, Section 5. Compare with the convergent series 1/1^1 - 1/2^2 + 1/3^3 - 1/4^4 + ... = Integral_{x = 0..1} x^x dx. See A083648.
More generally, for nonnegative integers a and b, the divergent series Sum_{n >= 0} (-1)^n*(a*n + b)^n is Borel summable to Integral_{x = 1..oo} x^(a-b-1)/x^(x^a) dx. (End)

cp's OEIS Frontend

A083649 Duplicate of A073009.

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A000312 a(n) = n^n; number of labeled mappings from n points to themselves (endofunctions).

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A083648 Decimal expansion of Sum_{n>=1} -(-1)^n/n^n = Integral_{x=0..1} x^x dx.

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A094289 Decimal expansion of Sum(1/p^p) where p is prime.

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A100085 Decimal expansion of Sum_{n>0} 1/(n!^n!).

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A099870 Decimal expansion of Sum_{n>0} 1/(n^log(n)).

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